Question

why do the maximum and minimum momenta have opposite signs? like cart 1 hitting cart 2 why would cart one be positive and cart to be negative?

Answer 1

see you have talked about the carts so im considering that you are talking about momentum. (AS we also have moment of force i.e torque).

See momentum is a vector quantity so the sigh in front of a no (like -5 or +3) just represent the direction with respect to each other. For example lets suppose you are having two carts which are moving in opposite direction with ,magnitude of momentum like 3 and 5 kgm/s. Now as they are moving in opposite direction you need to specify these vectors . let say Cart A (with value 3) is moving towards right and cart B (with value 5) moving towards left , now it depends on you or on your reference to take a direction +ve or -ve. you can define right direction as +ve the the vectors will be like

+3 and -5.

here - sign  indicates left direction is negative with respect to right or we can say that it makes an angle of 180 degree with right direction.

U can also define left direction to be +ve than the vectors will be

-3 and +5

here - sign  indicates right direction is negative with respect to left or we can say that it makes an angle of 180 degree with right direction.

so both are correct depending on your reference.

Now for why we prefer the 1st one , i.e right is +ve and left is -ve is because we already have a cartesian coordinate system which is this way so its quite convenient.

Hope you get what you are looking for. Thanku